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on June 11, 2017
for school
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on December 31, 2017
Since I haven't actually seen Sheldon's book yet, I can't really review the book itself. However, I agree very much with the point of view that the concept of the determinant, which usually gets defined early on as a complicated and rather bristly computation, has generally been taken to be too central to the development of the theoretical structure of linear spaces. I remember reading Hoffman & Kunze decades ago: It seemed like the first third of the book was dedicated to the behaviour and properties of this entity (the determinant) that had been defined essentially as a computation, without any clear motivation beneath it. For most of the way through H & K, I kept wondering, "When are they going to tell us what the heck this is about?"

So I agree with Sheldon's stated philosophy that the role of the determinant should be minimised until the STRUCTURE of the theory has begun to emerge, and then the DEFINITION will be something which arises more naturally as an expression of the structure of the theory; rather than the theory having to be explicated as a collection of properties which seem to be "accidental" consequences of an unmotivated computational definition. Indeed, if the theory is developed on the basis of the conceptual structures rather than on the computations, by the time you need the determinant, its actual definition is not such a big step. Whereas in the "traditional" approach, it shows up like a bull in a ring, unpredictable in nature, that has to be danced around while you try to figure out what to do with it.
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on March 3, 2004
As a graduate student in engineering at UC Berkeley, linear algebra is an essential tool for research and problem solving. I was lucky to take a linear algebra course taught from this book. This book is very concise, illustrative and very complete. None of the other popular books on linear algebra presents normed-vector spaces and their operators or the proof of the Jordan canonical form as precise and rigorous as this book. I think that Prof. S. Axler (the author) who is UC berkeley graduate himself has done a great job to write this book.
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on June 2, 2004
This is a short, elegant presentation of linear algebra appropriate for upper level undergraduate math majors with a theoretical bent. The student has perhaps taken a linear algebra course designed for engineers and scientists. Such a student is comfortable reading mathematics and writing proofs. It is meant to be read and re-read until the ideas are absorbed. The exercises are relatively easy and no answers are provided. With exercises of this sort you generally know if you are on the right track and they require you to understand the presentation in the text and process the ideas in a straight forward way.
Of course, there is nothing in this book about applications or the computational aspects of applying linear algebra.
The price is right. This could be a very useful purchase even if it's not assigned as a text.
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on April 30, 2004
I had this for an intermediate course on linear algebra (after the 1st course on abstract algebra) & I thought it was good for the level I was at. There's just enough stuff in this book to fill up a 1-term course, but no more so you'll have to get another book to find more applications, or other stuff to look at. The texts by Hoffman/Kunze or Finite-Dimensional Vector Spaces by Halmos are good references. As for this Axler book, I like how it's written in a relatively informal style, including comments in the margins by the author. I also like how he emphasizes the concepts of vector spaces, inner product spaces, etc rather than matrices (although they do appear but they're not emphasized) while determinants are done last. This is the only book I know of that does it this way & I think I liked it better like that.
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on March 27, 2002
I've only loooked through this book a bit, but I found the proofs to be very enlightening. It presents the ``correct'' view of linear algebra a the study of vectors spaces, not the study of R^n, and n x m matrices. The book introduces matrices towards the end for a very good reason: matrices aren't that important. The real substance of linear algebra: linear operators and vector spaces. Introducing linear operators as matrices would be like defining a homomorphism on a group by giving what the homomorphism does to the presentation for the group. An idiotic and counterintuitive method of defining homomorphisms. Yet in combinatorial group theory, it is helpful sometimes to do this. Much as it is sometimes helpful to work with matrices--but certainly not from the start.
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on January 30, 2015
The binding starts coming apart as soon as you open the book.
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on October 21, 2000
The text is preety terse and condense, it would be a delight if someone taking such courses from college and learned from an instructor. It would be a real bad problem if someone use the book for self study or as a reference material for more serious studies for no problems solutions were provided (this indirectly show the author definitely try to marketing the book to college instructors as teaching materials than as self-contained study materials. As a graduate student, I would stronly recommend users to consider Strang's Linear Algebra and its Application for their first choice as the real world approcah to problems and also as a stepping stone to more advanced theoretical studies in linear algebra.
For instructors this is an excellent choice for text book in the market for the price of the book is affordable to most students and meanwhile most important concepts and materials were hiding in the problem examples which will definitely challenging the most capable students and will be easy to curve the course grade in tests and exams.
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on August 28, 2002
Linear algebra is typically a course taught mostly to scientists and engineers who need to use the methods of undergraduate linear algebra to calculate things. To serve that need, and to avoid abstraction, the undergraduate linear algebra course teaches matrices, more matrices, and only matrices. Only years later do students discover that they can adapt those theorems about matrices to other situations. Often they do this quite carelessly, or if they are careful, avoid doing so and resort to more difficult methods. That is because they have not read this book. If every university had a class called "Linear Algebra Done Right", fewer math students would enter graduate school thinking that linear algebra is about matrices, and fewer otherwise sophisticated physics students would deal with linear operators by saying, "Dude, pretend it's a matrix!"
Intended to follow and complement that inescapable first class in linear algebra, "Linear Algebra Done Right" emphasizes the abstract over the concrete and elegance over brute force. Eigenvalues are banished to the latter half of the book and replaced with an abstract, definition-driven development of the basic theory. Students are liberated from coordinate hell and introduced to beautiful and powerful concepts. That perspective is unnecessary and probably confusing for engineers and scientists who only need the matrix methods for calculation, but it is a natural approach for mathematicians, physicists, and others who need a deep understanding of linear algebra to support their attack on more advanced mathematics.
A note on style: Although the material is introductory, the style is slightly more sophisticated than most introductory texts. To understand the material, the student must read closely, fill in the gaps "left as an exercise", and do the problems. This is the way all advanced mathematics books must be read, and "Linear Algebra Done Right" provides a gentle introduction to that manner of reading.
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on December 4, 2001
Sheldon Axler's "Linear Algebra Done Right" is an excellent book for the strong of heart. I am an undergraduate student majoring in mathematics, and my professors are obsessed with this book. I, however, am not.
First of all, there are no solutions to the exercises at the end of each chapter, so students are left frustrated when they cannot arrive at the next step of a proof.
Second, what the author sees and deems "obvious" as far as steps and recollections are concerned is not necessarily obvious to the reader.
Axler tries to motivate readers for the proofs by offering little exercises for them to "verify." That's overkill, but to many professors and analysts, overkill in the abstract is probably necessary in order to ensure a given student's success in an advanced linear algebra course.
I'm taking one such course to fulfill my requirements for a math major, and must say these abstract/proof courses get very monotonous and, thus, ridiculously boring. The text, itself, for this book does not particularly motivate me, and I expect to consult the chapters to learn and understand concepts, not to verify info from the chaper. Essentially, the flow of the concepts is ruined by the lack of examples; how are we supposed to verify ideas when the author hasn't really even exemplified the components of them yet?
I find myself falling asleep before I even complete one or two pages in this book. The layout is dull and the propositions and theorems seem endless.
My point is the following: That which is good for the instructor is not necessarily good for the student. Students need motivation, and it is difficult to achieve this goal without offering students detailed examples, interest-catchers and solutions to the time-consuming and overwhelming exercises and concepts.
In the realm of the college curriculum, this book is average. I understand the difficulty of making abstract algebra interesting, but this notion is precisely what students need. To the student math geniuses and professors that live and breathe math, this book is a gift from the gods.
To your average math student, however, that lacks patience and the desire to give up their free time to submerge himself/herself into this book, this book, like mine, will just end up sitting on the shelf and collecting dust.
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